Ergodic theory and the existence of strong ergodic limits

Abstract

In 1932, G.D. Birkhoff proved that, if (X,Sigma,v) is a finite measure space, f a summable function in X, and {U^lambda} a one-parameter group of measure preserving transformations on X, then for almost all x, the time average 1/Lambda integral from 0 to Lambda of f(U^lambda x)d lambda exists and is a summable function of x. In 1941, H. R. Pitt proved the important maximal ergodic theorem under the same hypothesis. This paper extends the pointwise and maximal ergodic theorems for groups of transformations to large measure spaces (theorems 3.12 and 3.5 respectively). Moreover, the maximal ergodic theorem yields as alternate proof of Lebesgue's theorem on differentiation of the indefinite integral of a summable function of a real variable. In 1951, Zygmund employed Pitt's maximal theorem to prove that membership in the class L log^+ Lis a sufficient condition to assure almost everywhere convergence of the pointwise ergodic limit with respect to two groups of measure preserving transformations. In theorem 4.10, we prove a refinement of Zygmund's L log^+ L result. Finally, a mean convergence theorem with respect to several groups of measure preserving transformations is obtained.